Cauchy Kernel of Slice Dirac Operator in Octonions with Complex Spine

Abstract

Holomorphic theory of one complex variable has a canonical extension to quaternions via the book structure in which the quaternions can be expressed as the union of complex planes through the common real axis. It is natural to consider extensions by replacing the $$\bar{\partial }$$ operator with the Dirac operator and by replacing the book structure of real spine with that of complex one. In this article, we introduce a new book structure for octonions which admits the complex plane as its spine instead of the real axis. The associated operator turns out to be the slice Dirac operator. In this setting of slice analysis, we establish the global Cauchy–Pompeiu integral formula.